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Properties

Label 12.0.6775409390765625.2
Degree $12$
Signature $[0, 6]$
Discriminant $6.775\times 10^{15}$
Root discriminant \(20.86\)
Ramified primes $3,5,29$
Class number $8$
Class group [8]
Galois group $D_6$ (as 12T3)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 2*x^10 - 17*x^9 + 23*x^8 + 115*x^7 + 10*x^6 - 279*x^5 - 168*x^4 + 90*x^3 + 171*x^2 + 189*x + 81)
 
Copy content gp:K = bnfinit(y^12 - 2*y^11 + 2*y^10 - 17*y^9 + 23*y^8 + 115*y^7 + 10*y^6 - 279*y^5 - 168*y^4 + 90*y^3 + 171*y^2 + 189*y + 81, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 2*x^10 - 17*x^9 + 23*x^8 + 115*x^7 + 10*x^6 - 279*x^5 - 168*x^4 + 90*x^3 + 171*x^2 + 189*x + 81);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 + 2*x^10 - 17*x^9 + 23*x^8 + 115*x^7 + 10*x^6 - 279*x^5 - 168*x^4 + 90*x^3 + 171*x^2 + 189*x + 81)
 

\( x^{12} - 2 x^{11} + 2 x^{10} - 17 x^{9} + 23 x^{8} + 115 x^{7} + 10 x^{6} - 279 x^{5} - 168 x^{4} + \cdots + 81 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $12$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[0, 6]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(6775409390765625\) \(\medspace = 3^{6}\cdot 5^{6}\cdot 29^{6}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(20.86\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}5^{1/2}29^{1/2}\approx 20.85665361461421$
Ramified primes:   \(3\), \(5\), \(29\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $D_6$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:  \(\Q(\sqrt{-15}, \sqrt{-87})\)

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{6}a^{7}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}$, $\frac{1}{18}a^{8}-\frac{1}{18}a^{7}-\frac{1}{9}a^{6}+\frac{4}{9}a^{5}+\frac{2}{9}a^{4}+\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{6}a-\frac{1}{2}$, $\frac{1}{18}a^{9}+\frac{1}{3}a^{5}+\frac{4}{9}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{54}a^{10}-\frac{1}{54}a^{9}+\frac{1}{54}a^{8}-\frac{2}{27}a^{7}-\frac{1}{27}a^{6}-\frac{5}{27}a^{5}+\frac{1}{9}a^{4}-\frac{1}{2}a^{3}+\frac{1}{6}a^{2}+\frac{1}{6}a$, $\frac{1}{138840588}a^{11}-\frac{749471}{138840588}a^{10}+\frac{2542049}{138840588}a^{9}-\frac{473365}{69420294}a^{8}+\frac{7674785}{138840588}a^{7}-\frac{8786131}{69420294}a^{6}-\frac{14723342}{34710147}a^{5}+\frac{148043}{5142244}a^{4}-\frac{1725827}{5142244}a^{3}+\frac{3616223}{15426732}a^{2}+\frac{879524}{3856683}a+\frac{762991}{5142244}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

Ideal class group:  $C_{8}$, which has order $8$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{8}$, which has order $8$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $5$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{2588777}{69420294}a^{11}-\frac{1224083}{11570049}a^{10}+\frac{3741691}{23140098}a^{9}-\frac{8941265}{11570049}a^{8}+\frac{35205259}{23140098}a^{7}+\frac{34878193}{11570049}a^{6}-\frac{71043475}{34710147}a^{5}-\frac{70244899}{7713366}a^{4}+\frac{8509580}{11570049}a^{3}+\frac{29169025}{7713366}a^{2}+\frac{19978778}{3856683}a+\frac{10782435}{2571122}$, $\frac{258541}{46280196}a^{11}-\frac{1972453}{138840588}a^{10}+\frac{1623583}{138840588}a^{9}-\frac{6844283}{69420294}a^{8}+\frac{24118093}{138840588}a^{7}+\frac{46431997}{69420294}a^{6}-\frac{11199698}{34710147}a^{5}-\frac{11037219}{5142244}a^{4}-\frac{65651545}{46280196}a^{3}+\frac{24737837}{15426732}a^{2}+\frac{2984804}{1285561}a+\frac{5931019}{5142244}$, $\frac{26263847}{138840588}a^{11}-\frac{23476625}{46280196}a^{10}+\frac{33563455}{46280196}a^{9}-\frac{85908803}{23140098}a^{8}+\frac{318432907}{46280196}a^{7}+\frac{394279579}{23140098}a^{6}-\frac{337019900}{34710147}a^{5}-\frac{2128634483}{46280196}a^{4}-\frac{15716567}{46280196}a^{3}+\frac{262149911}{15426732}a^{2}+\frac{77083645}{3856683}a+\frac{111404335}{5142244}$, $\frac{2087785}{15426732}a^{11}-\frac{2009073}{5142244}a^{10}+\frac{28335833}{46280196}a^{9}-\frac{32632297}{11570049}a^{8}+\frac{257514115}{46280196}a^{7}+\frac{249352615}{23140098}a^{6}-\frac{99266039}{11570049}a^{5}-\frac{1397480737}{46280196}a^{4}+\frac{240283109}{46280196}a^{3}+\frac{113786849}{15426732}a^{2}+\frac{118892789}{7713366}a+\frac{63927927}{5142244}$, $\frac{54751}{69420294}a^{11}-\frac{77527}{11570049}a^{10}+\frac{172376}{11570049}a^{9}-\frac{208919}{7713366}a^{8}+\frac{231711}{2571122}a^{7}-\frac{27283}{3856683}a^{6}-\frac{23630072}{34710147}a^{5}+\frac{14502863}{23140098}a^{4}+\frac{6976916}{11570049}a^{3}-\frac{1925971}{3856683}a^{2}+\frac{794635}{2571122}a-\frac{1070015}{2571122}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 901.086091143 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 901.086091143 \cdot 8}{2\cdot\sqrt{6775409390765625}}\cr\approx \mathstrut & 2.69424891561 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 + 2*x^10 - 17*x^9 + 23*x^8 + 115*x^7 + 10*x^6 - 279*x^5 - 168*x^4 + 90*x^3 + 171*x^2 + 189*x + 81) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^12 - 2*x^11 + 2*x^10 - 17*x^9 + 23*x^8 + 115*x^7 + 10*x^6 - 279*x^5 - 168*x^4 + 90*x^3 + 171*x^2 + 189*x + 81, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 2*x^10 - 17*x^9 + 23*x^8 + 115*x^7 + 10*x^6 - 279*x^5 - 168*x^4 + 90*x^3 + 171*x^2 + 189*x + 81); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 2*x^11 + 2*x^10 - 17*x^9 + 23*x^8 + 115*x^7 + 10*x^6 - 279*x^5 - 168*x^4 + 90*x^3 + 171*x^2 + 189*x + 81); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_6$ (as 12T3):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 12
The 6 conjugacy class representatives for $D_6$
Character table for $D_6$

Intermediate fields

\(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{145}) \), 3.1.87.1 x3, \(\Q(\sqrt{-15}, \sqrt{-87})\), 6.0.658503.1, 6.0.2838375.1 x3, 6.2.27437625.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 6 siblings: 6.0.2838375.1, 6.2.27437625.1
Minimal sibling: 6.0.2838375.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{4}$ R R ${\href{/padicField/7.6.0.1}{6} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{2}$ ${\href{/padicField/17.3.0.1}{3} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{6}$ ${\href{/padicField/23.2.0.1}{2} }^{6}$ R ${\href{/padicField/31.2.0.1}{2} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.2.0.1}{2} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(29\) Copy content Toggle raw display 29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
29.2.2.2a1.2$x^{4} + 48 x^{3} + 580 x^{2} + 96 x + 33$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)